Spin–phonon coupling and thermodynamic behaviour in YCrO₃ and LaCrO₃: inelastic neutron scattering and lattice dynamics

We have measured the INS spectra of YCrO₃ and LaCrO₃ (figure 2) up to 550 K. The temperature range of measurements includes the magnetic transition in both the compounds (T_N (YCrO₃) [27] ∼140 K and T_N (LaCrO₃) [52] ∼291 K). Since both the compounds are magnetic, the INS data contain both phonon and magnetic contributions. The magnetic contribution in the INS spectrum is of two kinds, one is due to well-defined magnon excitations that dominates below the magnetic transition temperature, and other is the quasielastic scattering from paramagnetically rotating spins that dominates above the transition temperature. It is known that the magnetic contribution to the INS spectrum dominates at low momentum transfers (Q) due to the magnetic form factor. Therefore, two Q-domains were considered; i.e., high-Q (4–7 Å⁻¹) and low-Q (1–4 Å⁻¹) in order to extract the magnetic contribution in the INS data at low-Q and the phonon contribution at high-Q.

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The temperature dependence of the Bose-factor corrected S(Q, E) plots of YCrO₃ and LaCrO₃ are shown in figure 2. At low temperatures (up to 315 K), the low-Q data show a larger elastic line as compared to the high-Q data. We could see that the energy spectrum close to the elastic line is also very sensitive to Q, i.e. at lower Q (1–4 Å⁻¹), there is larger intensity in the low energy spectrum and at higher momentum transfer (Q ∼ 4–7 Å⁻¹), it reduces significantly. As noted above, the magnetic signal is more pronounced at low Q, and vanishes at higher Q, following the magnetic form factor. We speculate that this quasi-elastic scattering originates from paramagnetic spin fluctuations.

The low-Q inelastic spectrum in YCrO₃ shows significant intensity at energy around 20 meV at 100 K (below magnetic transition temperature T_N ∼ 140 K), which decays rapidly above T_N . This indicates that there is significant contribution by magnon excitations at about 20 meV. Further, the intensity around zero energy is contributed by paramagnetic scattering of neutrons by Cr ions. The inelastic spectrum averaged over high Q does not show any prominent change with temperature. However, in case of LaCrO₃, we do not observe any significant difference between the low- and high-Q data at 175 K (T_N ∼ 291 K) (figure 2), and so, the contribution from the magnetic excitations seems to be insignificant.

Our INS data on polycrystalline sample of LaCrO₃ are insufficient to extract the magnon contribution quantitatively. Hence, we could not perform any modeling to derive inferences about the magnetic interaction, which could be better obtained from magnon dispersion data from single crystal measurements. To explain the major differences in the measured low-energy INS spectrum at low-Q, we have performed magnon dispersion simulation using spinw software [51]. We have assumed the G-type AFM structure of both the compounds. The magnetic exchange interaction parameters (J values) between various Cr³⁺ atoms are taken from available literature [53, 54]. The J₁ value for YCrO₃ (J₁ = −1.685 meV and J₂ = −0.428 meV) [53] is much smaller in comparison to that for LaCrO₃ (J₁ = −8.8 meV and J₂ = −0.38 meV) [54]. The large difference between the exchange parameters is also reflected in the difference in their T_N values, which are ∼140 K for YCrO₃ [27] and ∼291 K for LaCrO₃ [52]. The calculated magnon dispersion relation using these parameters are shown in figure 3(a). The calculations show a flat magnon branch at ∼10 meV and ∼90 meV in YCrO₃ and LaCrO₃ respectively, which will contribute a large magnon density of states and show up in INS measurements at 10 meV and 90 meV respectively. Our INS measurements of YCrO₃ at low-Q show (figure 2) a prominent peak at energy around ∼20 meV at 100 K (below the magnetic transition), which is qualitatively consistent with the above theoretical prediction of large magnon contribution in YCrO₃ (figure 3(a)). However, for LaCrO₃ the predicted peak magnon contribution at 90 meV. Such high energy excitations cannot be observed at low Q values, and would require a very high Q value of about 7 Å⁻¹. The magnetic form factor falls rapidly at higher momentum transfer. As shown in figure 3(c), the magnetic form factor for Cr³⁺ at 7 Å⁻¹ falls to 6%, which makes it difficult for magnons to be seen in the INS experiments performed using a polycrystalline sample. This might be the reason that in LaCrO₃, we do not see significant variation in the low-Q intensity in the spectra across the magnetic transition as we do see in YCrO₃.

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As mentioned above, the neutron inelastic scattering spectrum from powder sample contains rich information of dynamics of atoms from the entire Brillouin zone. For magnetic systems, it also contains the magnon density of states. McQueeney et al [55] have shown that the information of magnetic structure is embedded in magnon density of states and can be extracted from the powder neutron inelastic scattering data. Using the Heisenberg Hamiltonian, the authors derive the magnon dispersion relation and show that for different magnetic structure, the magnon density of states exhibits distinct features in the von Hove singularities irrespective of the magnetic exchange interaction strength. For, a three-dimension FM (F-type) cubic perovskite, the magnon density of states shows five von-Hove singularities contributed by different high symmetry zone boundary modes. The A-type antiferromagnetic cubic perovskite will have many von-Hove singularities, while the C-type AFM structure will have only three von Hove singularity features in magnon density of states. The authors found that the G-type magnon density of states show very distinct magnon density of states and it contains only one cusp like singularity near the maximum magnon energy.

In the inelastic experiment, we measure the dynamical structure factor S(Q, E), which contains the information of both the magnon and phonon density of states. Hence, the dynamical structure factor contributed from the magnon dynamics would also exhibit distinct von-Hove singularities as described above. To probe these features in our inelastic neutron spectrum, we have further processed our data to extract the magnon structure factor S_mag(Q, E) by subtracting the high-Q inelastic neutron spectrum from the low-Q inelastic spectrum. We obtained the S_mag(Q, E) for YCrO₃ (figure 3(b)). The magnon dynamical structure factor consists of one von Hove singularity at around ∼20 meV energy, which is a strong signature of the G-type AFM structure. The conclusion that YCrO₃ compound establishes a G-type AFM is based on the shape of the magnon peak. McQueeney et al [55] showed that the information of magnetic structure is embedded in magnon density of states and may be extracted from the powder neutron inelastic scattering data. This result supplements the previous diffraction observation [27] and justifies our calculation with the G-type AFM structure.

Here, our aim was to describe the richness of the information contained in the INS spectrum, which can be further utilized to characterize the material properties. In order to confirm that the peak observed at around 20 meV in S_mag(Q, E) has magnetic origin and hence should follow the magnetic form factor of associated magnetic ion, we have plotted (figure 3(c)). This figure gives the integrated intensity in the S_mag(Q, E) over 17–22 meV as a function of momentum transfer from 1–7 Å⁻¹. We have also calculated the Q dependence of magnetic form factor of Cr⁺³ using analytical method [56] and compared it with the variation of the integrated peak intensity (S_mag(Q, E)) with Q. We found a very good agreement between the two, which unambiguously confirms that the peak ∼20 meV in S_mag(Q, E) is contributed by magnetic excitation (magnons).

Further, the spectra derived from S(Q, E) data using equation (2), within the incoherent approximation in both the Q regimes, are shown in figure 4. We find that for both the compound the low-Q data show large variation in the intensity as a function of temperature. However, for the high-Q data we do not observe any significant change in the spectra except at the lowest temperature. Further, in the low-Q data of YCrO₃ at 100 K, which is below the magnetic transition temperature (∼140 K), there is a large intensity of the low energy inelastic spectra (∼20 meV) as compared to the data collected at higher temperatures. This is expected to be due to a strong magnetic signal. At 175 K, it is found that there is considerable decrease of the intensity of the low energy peaks around 20 meV indicating loss of the magnetic signal.

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We have calculated the relaxed structure for both the compounds in the non-magnetic as well as magnetic configurations, and the results are given in table 1. Without the magnetism the a-lattice constant is significantly overestimated by about 5%. We find that the magnetism substantially reduces the overestimate to about 2%. Further, in order to see the effect of the direction of magnetic moment on the structure we performed both collinear and non-collinear calculations (moment along a-axis) for YCrO₃. We find that the relaxed lattice parameters are quite similar in the two cases and differ from the experiments by about 2% (table 1), which is acceptable in standard DFT calculation. The calculated magnetic moment of Cr ion in both the compounds is found to be 3.0 μ_B, which is due to the fact that Cr atom occurs in +3 ionic states in the compounds and all the remaining d³ electron occupy the t_3g orbital.

It may be noted that above the magnetic transition temperature the compounds become paramagnetic, where the magnetic ordering is lost which results in marginally reduced effective interaction between magnetic ions. Hence, in order to reproduce the phonon spectrum in paramagnetic phase we cannot use nonmagnetic calculation (magnetic moment of Cr = 0 μ_B, which is equivalent to magnetic quenching). The calculation of phonon spectrum in paramagnetic phase can be obtained by calculating the phonon spectrum over a large set of randomly oriented spin configurations which is not a feasible option but may be in near future one can use machine learning methods to achieve the same. However, in the present case, we can estimate the changes in the phonon spectrum between few magnetic configurations and can have a sense of change in phonon energies across the magnetic transition and this could be useful to identify modes which may show strong spin–phonon coupling.

The calculated partial density of states of various atoms in collinear and noncollinear magnetic configuration for YCrO₃ are shown in figure 5. It may be noted that the low energy phonon spectrum below 20 meV does not show any change, while small changes are observed at higher energies (above 45 meV) that are mostly contributed by chromium and oxygen atoms. This can be understood from the fact that the magnetic interaction between chromium atoms are mediated by oxygens and change in the direction and magnitude of magnetic moment will affect the Cr–O bond length and O–Cr–O bond angle, and hence the phonons. It has been reported that the Raman modes [32] in YCrO₃ around ∼560 cm⁻¹ (∼70 meV) show significant changes across the magnetic transition temperature, and therefore are expected to be influenced by magnetic interactions, which is consistent with the above ab initio results.

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The phonon density of states of LaCrO₃ and YCrO₃ differ significantly (figure 5) below 50 meV, which are dominated by Y and La dynamics. The differences are mainly attributed to difference in various bond lengths and strengths in both the compounds. However, the high energy phonon modes above 50 meV, which mainly arise from Cr–O stretching, do not change much since in the both the compound the Cr–O bond length is nearly same.

We compare in figure 6 the INS measurements at T = 310 K with phonon calculations that includes the magnetic interaction (spin polarized phonon calculation). The phonon density of states obtained from measurements is neutron weighted. Hence, in order to compare with the data, we have done the neutron weighting of the calculated phonon data using equation (2). The calculated phonon density of states for YCrO₃ shows good agreement with the measurements. We observe slightly underestimated phonon energies with respect to measurements. That could be understood better in terms of overestimated lattice parameters or under-binding effect of GGA exchange correlation function in DFT approach. In case of LaCrO₃, the calculated phonon density of states is found to be in good agreement with the neutron data below 55 meV, while there is significant underestimation of high energy phonon density of states. The significant underestimation of high energy modes might be attributed to strong correlation effect of La deep-d orbital electrons, which is not very well accounted in DFT methods. However, qualitatively overall spectrum matches very well with the measurements.

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Further, we have shown (figure 6) the contribution of individual atoms to the neutron inelastic scattering spectrum. We found that for both the compounds the contribution from oxygen atoms is dominated across the entire spectral range. The Cr and R (=Y, La) atomic dynamics are significantly contributed in the spectrum below 70 meV and 30 meV respectively.

The phonon density of states is an integrated spectrum of phonons in the entire Brillouin zone. To give a sense of quantitative changes in specific phonon modes due to collinear and non-collinear magnetic configuration in YCrO₃, we have shown the calculated zone-center phonon frequencies in table 2 and figure 7 in the G-type magnetic configuration in two type of calculations, i.e., non-collinear vs collinear. Group theoretical analysis of Pnma symmetry of RCO₃ (R = Y, La) shows that at zone-center the total of 60 normal modes can be classified as follows:

$$\begin{align*}\hfill {\Gamma} :7{\mathrm{A}}_{\mathrm{g}}& +5{\mathrm{B}}_{1\mathrm{g}}+7{\mathrm{B}}_{2\mathrm{g}}+5{\mathrm{B}}_{3\mathrm{g}}+8{\mathrm{A}}_{1\mathrm{u}}+10{\mathrm{B}}_{1\mathrm{u}}\hfill \\ \hfill & +8{\mathrm{B}}_{2\mathrm{u}}+10{\mathrm{B}}_{3\mathrm{u}}.\hfill \end{align*}$$

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Here, we have 24 Raman active modes and 36 are infrared active. One may note that a few zone-center phonon modes show significant change with magnetic configuration, indicating that these modes exhibit strong spin–phonon coupling. We find that most of the zone-center modes in YCrO₃ above 45 meV show significantly change in phonon energies (∼5%) between non-collinear and collinear calculations. This also reflects strongly in the density of states calculations shown in figure 5. Further, we have shown the calculated zone-center phonon modes of LaCrO₃ in table 2. We may observe that most of the zone-center phonon modes in LaCrO₃ are lower in energy than that in YCrO₃, which might be due to larger La mass and weaker bonding between La and O. Experimental data of the zone-center modes for YCrO₃ and LaCrO₃ available from references [20, 25, 57, 58] respectively, are in fair agreement with our ab initio calculations. We have used both the supercell scheme and DFPT method for calculation of the zone-center modes. The DFPT calculation for collinear cases of YCrO₃ and LaCrO₃ are found to be in excellent agreement with that using the supercell scheme (table 2).

In figure 8, we have shown the eigenvectors of some of the representative modes in YCrO₃, which show significant change in phonon energy with magnetic configurations. The most striking feature of these modes is that these modes are mainly dominated by oxygen dynamics. The dynamics of oxygen will cause variations in the O–Cr bond and O–Cr–O bond angle, which are critical for antiferromagnetic interaction. Hence, different types of magnetic configurations result in difference in strength of the force constants for these bonds and bond angles, which in turn reflect in changes in the phonon frequencies. As the stretching and bending modes in these compounds occur at high energies, we expect that significant changes in the high energy Raman and IR modes would be observed in presence of strong magnetic field.

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The calculated elastic constants and elastic compliances in YCrO₃ and LaCrO₃ are given in table 3. The calculated elastic constant C₂₂ in YCrO₃ shows significantly large difference between the non-collinear and collinear calculations. Neutron powder diffraction has been used to study the high temperature behaviour of RCrO₃ (R = Y, La). An orthorhombic to a rhombohedral structural phase transition has been reported [59] in LaCrO₃ at about 533 K, while in YCrO₃ measurements show [60] no transition up to 1200 K. Detailed experimental data on anisotropic thermal expansion behaviour are available for YCrO₃, but not for LaCrO₃. Here we only report the thermal expansion calculation for YCrO₃ using collinear calculation in a G-type magnetic structure. Previously our thermal expansion calculations using collinear magnetic configuration in YMnO₃ [61] and GaFeO₃ [62] were found to be in good agreement with the available experimental data.

It is important to mention here that a large number of phonons in the entire Brillouin zone contribute [44, 48, 63] to the thermal expansion behaviour. The calculated anisotropic Grüneisen parameters (Γ(E)) averaged over the Brillouin zone are shown in figure 9. We note that Γ(E) have large values for energies below 10 meV. However, the density of states below 10 meV is relatively small, and so it does not contribute much to the thermal expansion around 300 K and above. At higher energies above 10 meV, the values of all the three anisotropic Grüneisen parameters (Γ_a , Γ_b and Γ_c ) in the entire energy range are found to be nearly same, except that Γ_a is negative around 20 meV. To get a numerical sense of the dependence of the Grüneisen parameters on the linear thermal expansion coefficients for YCrO₃, we re-express the equation (3) using the elastic compliance values (table 3) as follows:

$$\begin{equation}{\alpha }_{a}\left(T\right)=\frac{1}{{V}_{0}}\sum _{q,j}{C}_{{l}^{\prime }}\left(q,j,T\right)\left[4.4{{\Gamma}}_{a}-1.2{{\Gamma}}_{b}-1.7{{\Gamma}}_{c}\right]\end{equation} \tag{ 6 }$$

$$\begin{equation}{\alpha }_{b}\left(T\right)=\frac{1}{{V}_{0}}\sum _{q,j}{C}_{{l}^{\prime }}\left(q,j,T\right)\left[-1.2{{\Gamma}}_{a}+4.4{{\Gamma}}_{b}-1.1{{\Gamma}}_{c}\right]\end{equation} \tag{ 7 }$$

$$\begin{equation}{\alpha }_{c}\left(T\right)=\frac{1}{{V}_{0}}\sum _{q,j}{C}_{{l}^{\prime }}\left(q,j,T\right)\left[-1.7{{\Gamma}}_{a}-1.1{{\Gamma}}_{b}+4.7{{\Gamma}}_{c}\right]\end{equation} \tag{ 8 }$$

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It can be seen that the anisotropic Grüneisen parameters (Γ_a, Γ_b and Γ_c ) as shown in figure 9 would lead to anisotropic linear thermal expansion coefficients. The linear thermal expansion coefficients α_b would have maximum positive value, while α_a would have minimum positive value. The calculated linear thermal expansion coefficients for YCrO₃ in the orthorhombic phase above 140 K (paramagnetic phase), as a function of temperature are shown in figure 10(a). In figure 10(b), we have compared the calculated values with the available experimental data [60] on YCrO₃. The experimental observation of anisotropic thermal expansion behaviour of YCrO₃ in the paramagnetic phase above 140 K is very well reproduced (figure 10(b)) by our calculation.

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Zhu et al [60] inferred an isosymmetric structural phase transition at 900 K from an anomaly in the structural parameters. The information about isostructural transition is difficult to get in powder INS data. We also note that all the DFT calculations are performed at T = 0; hence, we could not comment on the anomaly at 900 K from our experiments and calculations. Our calculations of the changes in the lattice parameters with temperature are compared with the experimental data in figure 10(b), which indicate an overall good agreement. However, the anomaly in the experimental data, which appears rather small in figure 10(b), is not seen in the calculation.

To investigate the specific anharmonic phonon energies responsible for the thermal expansion behaviour, we have calculated the contribution to the linear thermal expansion coefficients (figure 11(a)) from phonons of various energies averaged over the Brillouin zone at 300 K in YCrO₃. We find that the phonons of energy around 10–20 meV (figure 9) contribute maximum to the thermal expansion along all the three axes. The calculated partial phonon density of states (figure 5) show that at these energies the contribution from Y atoms is more in comparison to that of Cr and O atoms.

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To understand the nature of these phonons we have calculated (figure 11(b)) the contribution of phonons of energy E to the mean-squared amplitude [48] of atoms at 300 K, which shows that at energies around 10–20 meV the u² values of Y atoms are significantly larger compared to Cr and O atoms. The modes corresponding to rotation of CrO₆ and YO₈ units are found at higher energies above 50 meV and 30 meV respectively.